A002133 -id:A002133 - cp's OEIS frontend

A216669 Total number of parts in all partitions of n into 2 sizes of parts (A002133).

2, 5, 14, 20, 39, 52, 74, 102, 134, 158, 208, 259, 284, 361, 409, 488, 538, 634, 678, 838, 857, 1006, 1038, 1270, 1264, 1495, 1500, 1776, 1761, 2084, 2062, 2443, 2300, 2795, 2680, 3194, 3076, 3544, 3403, 4080, 3804, 4518, 4282, 5037, 4673, 5626, 5127, 6088

Offset: 3

Geoffrey Critzer, Sep 13 2012

nonn

Also column k=2 of A255768. - Omar E. Pol, Jul 26 2015

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Cf. A002133, A216665, A255768.

nn=40;ss=Sum[Sum[y^2 x^(i+j)/(1-y x^i)/(1-y x^j),{j,1,i-1}], {i,1,nn}]; CoefficientList[Series[D[ss,y]/.y->1,{x,0,nn}],x]

a(n) = Sum_{k=2..n-1} k * A216665(n,k).

A045845 Duplicate of A002133.

Original entry on oeis.org

1, 2, 5, 6, 11, 13, 17, 22, 27, 29, 37, 44, 44, 55, 59, 68, 71, 81, 82, 102, 97, 112, 109

Offset: 3

dead

A238279 Triangle read by rows: T(n,k) is the number of compositions of n into nonzero parts with k parts directly followed by a different part, n>=0, 0<=k<=A004523(n-1).

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 1, 2, 10, 4, 4, 12, 14, 2, 2, 22, 29, 10, 1, 4, 26, 56, 36, 6, 3, 34, 100, 86, 31, 2, 4, 44, 148, 200, 99, 16, 1, 2, 54, 230, 374, 278, 78, 8, 6, 58, 322, 680, 654, 274, 52, 2, 2, 74, 446, 1122, 1390, 814, 225, 22, 1, 4, 88, 573, 1796, 2714, 2058, 813, 136, 10, 4, 88, 778, 2694, 4927

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 22 2014

Same as A238130, with zeros omitted.
Last elements in rows are 1, 1, 2, 2, 1, 4, 2, 1, 6, 2, 1, 8, ... with g.f. -(x^6+x^4-2*x^2-x-1)/(x^6-2*x^3+1).
For n > 0, also the number of compositions of n with k + 1 runs. - Gus Wiseman, Apr 10 2020

			Triangle starts:
  00:  1;
  01:  1;
  02:  2;
  03:  2,   2;
  04:  3,   4,   1;
  05:  2,  10,   4;
  06:  4,  12,  14,    2;
  07:  2,  22,  29,   10,    1;
  08:  4,  26,  56,   36,    6;
  09:  3,  34, 100,   86,   31,    2;
  10:  4,  44, 148,  200,   99,   16,    1;
  11:  2,  54, 230,  374,  278,   78,    8;
  12:  6,  58, 322,  680,  654,  274,   52,    2;
  13:  2,  74, 446, 1122, 1390,  814,  225,   22,   1;
  14:  4,  88, 573, 1796, 2714, 2058,  813,  136,  10;
  15:  4,  88, 778, 2694, 4927, 4752, 2444,  618,  77,  2;
  16:  5, 110, 953, 3954, 8531, 9930, 6563, 2278, 415, 28, 1;
  ...
Row n=5 is 2, 10, 4 because in the 16 compositions of 5
  ##:  [composition]  no. of changes
  01:  [ 1 1 1 1 1 ]   0
  02:  [ 1 1 1 2 ]   1
  03:  [ 1 1 2 1 ]   2
  04:  [ 1 1 3 ]   1
  05:  [ 1 2 1 1 ]   2
  06:  [ 1 2 2 ]   1
  07:  [ 1 3 1 ]   2
  08:  [ 1 4 ]   1
  09:  [ 2 1 1 1 ]   1
  10:  [ 2 1 2 ]   2
  11:  [ 2 2 1 ]   1
  12:  [ 2 3 ]   1
  13:  [ 3 1 1 ]   1
  14:  [ 3 2 ]   1
  15:  [ 4 1 ]   1
  16:  [ 5 ]   0
there are 2 with no changes, 10 with one change, and 4 with two changes.

Joerg Arndt and Alois P. Heinz, Rows n = 0..180, flattened

Columns k=0-10 give: A000005 (for n>0), 2*A002133, A244714, A244715, A244716, A244717, A244718, A244719, A244720, A244721, A244722.
Row lengths are A004523.
Row sums are A011782.
The version counting adjacent equal parts is A106356.
The version for ascents/descents is A238343.
The version for weak ascents/descents is A333213.
The k-th composition in standard-order has A124762(k) adjacent equal parts, A124767(k) maximal runs, A333382(k) adjacent unequal parts, and A333381(k) maximal anti-runs.
Cf. A064113, A333214, A333216.

b:= proc(n, v) option remember; `if`(n=0, 1, expand(
      add(b(n-i, i)*`if`(v=0 or v=i, 1, x), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..14);

b[n_, v_] := b[n, v] = If[n == 0, 1, Expand[Sum[b[n-i, i]*If[v == 0 || v == i, 1, x], {i, 1, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Maple *)
Table[If[n==0,1,Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Split[#]]==k+1&]]],{n,0,12},{k,0,If[n==0,0,Floor[2*(n-1)/3]]}] (* Gus Wiseman, Apr 10 2020 *)

T_xy(max_row) = {my(N=max_row+1, x='x+O('x^N),h=(1+ sum(i=1,N,(x^i-y*x^i)/(1+y*x^i-x^i)))/(1-sum(i=1,N, y*x^i/(1+y*x^i-x^i)))); for(n=0,N-1, print(Vecrev(polcoeff(h,n))))}
T_xy(16) \\ John Tyler Rascoe, Jul 10 2024

G.f.: A(x,y) = ( 1 + Sum_{i>0} ((x^i)*(1 - y)/(1 + y*x^i - x^i)) )/( 1 - Sum_{i>0} ((y*x^i)/(1 + y*x^i - x^i)) ). - John Tyler Rascoe, Jul 10 2024

A116608 Triangle read by rows: T(n,k) is number of partitions of n having k distinct parts (n>=1, k>=1).

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 2, 5, 4, 6, 1, 2, 11, 2, 4, 13, 5, 3, 17, 10, 4, 22, 15, 1, 2, 27, 25, 2, 6, 29, 37, 5, 2, 37, 52, 10, 4, 44, 67, 20, 4, 44, 97, 30, 1, 5, 55, 117, 52, 2, 2, 59, 154, 77, 5, 6, 68, 184, 117, 10, 2, 71, 235, 162, 20, 6, 81, 277, 227, 36, 4, 82, 338, 309, 58, 1

Offset: 1

Emeric Deutsch, Feb 19 2006

Row n has floor([sqrt(1+8n)-1]/2) terms (number of terms increases by one at each triangular number).
Row sums yield the partition numbers (A000041).
Row n has length A003056(n), hence the first element of column k is in row A000217(k). - Omar E. Pol, Jan 19 2014

			T(6,2) = 6 because we have [5,1], [4,2], [4,1,1], [3,1,1,1], [2,2,1,1] and [2,1,1,1,1,1] ([6], [3,3], [3,2,1], [2,2,2] and [1,1,1,1,1,1] do not qualify).
Triangle starts:
  1;
  2;
  2,  1;
  3,  2;
  2,  5;
  4,  6, 1;
  2, 11, 2;
  4, 13, 5;
  3, 17, 10;
  4, 22, 15, 1;
  ...

Alois P. Heinz, Rows n = 1..500, flattened
Emmanuel Briand, On partitions with k corners not containing the staircase with one more corner, arXiv:2004.13180 [math.CO], 2020.
Sang June Lee and Jun Seok Oh, On zero-sum free sequences contained in random subsets of finite cyclic groups, arXiv:2003.02511 [math.CO], 2020.

Cf. A000041, A000005, A000070, A002133, A002134.
Cf. A060177 (reflected rows). - Alois P. Heinz, Jan 29 2014
Cf. A274174.

g:=product(1+t*x^j/(1-x^j),j=1..30)-1: gser:=simplify(series(g,x=0,27)): for n from 1 to 23 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 23 do seq(coeff(P[n],t^j),j=1..floor(sqrt(1+8*n)/2-1/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 else []; for j from 0 to n/i do zip((x, y)
      ->x+y, %, [`if`(j>0, 0, [][]), b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> subsop(1=NULL, [b(n, n)])[]:
seq(T(n), n=1..30); # Alois P. Heinz, Nov 07 2012
# third program
nDiffParts := proc(L)
        nops(convert(L,set)) ;
end proc:
A116608 := proc(n,k)
        local a,L;
        a :=0 ;
        for L in combinat[partition](n) do
                if nDiffParts(L) = k then
                        a := a+1 ;
                end  if;
        end do:
        a ;
end proc: # R. J. Mathar, Jun 07 2024

p=Product[1+(y x^i)/(1-x^i),{i,1,20}];f[list_]:=Select[list,#>0&];Flatten[Map[f,Drop[CoefficientList[Series[p,{x,0,20}],{x,y}],1]]] (* Geoffrey Critzer, Nov 28 2011 *)
Table[Length /@ Split[Sort[Length /@ Union /@ IntegerPartitions@n]], {n, 22}] // Flatten (* Robert Price, Jun 13 2020 *)

from math import isqrt
from itertools import count, islice
from sympy.utilities.iterables import partitions
def A116608_gen(): # generator of terms
    return (sum(1 for p in partitions(n) if len(p)==k) for n in count(1) for k in range(1,(isqrt((n<<3)+1)-1>>1)+1))
A116608_list = list(islice(A116608_gen(),30)) # Chai Wah Wu, Sep 14 2023

from functools import cache
@cache
def P(n: int, k: int, r: int) -> int:
    if n == 0: return 1 if k == 0 else 0
    if k == 0: return 0
    if r == 0: return 0
    return sum(P(n - r * j, k - 1, r - 1)
               for j in range(1, n // r + 1)) + P(n, k, r - 1)
def A116608triangle(rows: int) -> list[int]:
    return list(filter(None, [P(n, k, n) for n in range(1, rows)
                              for k in range(1, n + 1)]))
print(A116608triangle(22)) # Peter Luschny, Sep 14 2023, courtesy of Amir Livne Bar-on

G.f.: -1 + Product_{j=1..infinity} 1 + tx^j/(1-x^j).
T(n,1) = A000005(n) (number of divisors of n).
T(n,2) = A002133(n).
T(n,3) = A002134(n).
Sum_{k>=1} k * T(n,k) = A000070(n-1).
Sum_{k>=0} k! * T(n,k) = A274174(n). - Alois P. Heinz, Jun 13 2016
T(n + A000217(k), k) = A000712(n), for 0 <= n <= k [Briand]. - Álvar Ibeas, Nov 04 2020

A238130 Triangle read by rows: T(n,k) is the number of compositions into nonzero parts with k parts directly followed by a different part, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 2, 2, 0, 0, 3, 4, 1, 0, 0, 2, 10, 4, 0, 0, 0, 4, 12, 14, 2, 0, 0, 0, 2, 22, 29, 10, 1, 0, 0, 0, 4, 26, 56, 36, 6, 0, 0, 0, 0, 3, 34, 100, 86, 31, 2, 0, 0, 0, 0, 4, 44, 148, 200, 99, 16, 1, 0, 0, 0, 0, 2, 54, 230, 374, 278, 78, 8, 0, 0, 0, 0, 0, 6, 58, 322, 680, 654, 274, 52, 2, 0, 0, 0, 0, 0

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 21 2014

First column (k=0) is A000005, second column (k=1) is 2*A002133.

Row sums are A011782.

			Triangle starts:
00:  1,
01:  1, 0,
02:  2, 0, 0,
03:  2, 2, 0, 0,
04:  3, 4, 1, 0, 0,
05:  2, 10, 4, 0, 0, 0,
06:  4, 12, 14, 2, 0, 0, 0,
07:  2, 22, 29, 10, 1, 0, 0, 0,
08:  4, 26, 56, 36, 6, 0, 0, 0, 0,
09:  3, 34, 100, 86, 31, 2, 0, 0, 0, 0,
10:  4, 44, 148, 200, 99, 16, 1, 0, 0, 0, 0,
11:  2, 54, 230, 374, 278, 78, 8, 0, 0, 0, 0, 0,
12:  6, 58, 322, 680, 654, 274, 52, 2, 0, 0, 0, 0, 0,
13:  2, 74, 446, 1122, 1390, 814, 225, 22, 1, 0, 0, 0, 0, 0,
...
Row 5 is [2, 10, 4, 0, 0, 0] because in the 16 compositions of 5
##:  [composition]  no. of changes
01:  [ 1 1 1 1 1 ]   0
02:  [ 1 1 1 2 ]   1
03:  [ 1 1 2 1 ]   2
04:  [ 1 1 3 ]   1
05:  [ 1 2 1 1 ]   2
06:  [ 1 2 2 ]   1
07:  [ 1 3 1 ]   2
08:  [ 1 4 ]   1
09:  [ 2 1 1 1 ]   1
10:  [ 2 1 2 ]   2
11:  [ 2 2 1 ]   1
12:  [ 2 3 ]   1
13:  [ 3 1 1 ]   1
14:  [ 3 2 ]   1
15:  [ 4 1 ]   1
16:  [ 5 ]   0
there are 2 with no changes, 10 with one change, and 4 with two changes.

Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Cf. A238279 (same sequence with zeros omitted).
Cf. A106356 (compositions with k successive parts same).
Cf. A225084 (compositions with maximal up-step k).

b:= proc(n, v) option remember; `if`(n=0, 1, expand(
      add(b(n-i, i)*`if`(v=0 or v=i, 1, x), i=1..n)))
    end:
T:= n-> seq(coeff(b(n, 0), x, i), i=0..n):
seq(T(n), n=0..14);

b[n_, v_] := b[n, v] = If[n == 0, 1, Sum[b[n-i, i]*If[v == 0 || v == i, 1, x], {i, 1, n}]]; T[n_] := Table[Coefficient[b[n, 0], x, i], {i, 0, n}]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 12 2015, translated from Maple *)

A060177 Triangle of generalized sum of divisors function, read by rows.

Original entry on oeis.org

1, 2, 1, 2, 2, 3, 5, 2, 1, 6, 4, 2, 11, 2, 5, 13, 4, 10, 17, 3, 1, 15, 22, 4, 2, 25, 27, 2, 5, 37, 29, 6, 10, 52, 37, 2, 20, 67, 44, 4, 1, 30, 97, 44, 4, 2, 52, 117, 55, 5, 5, 77, 154, 59, 2, 10, 117, 184, 68, 6, 20, 162, 235, 71, 2, 36, 227, 277, 81, 6, 1, 58, 309, 338

Offset: 1

N. J. A. Sloane, Mar 20 2001

Lengths of rows are 1 1 2 2 2 3 3 3 3 4 4 4 4 4 ... (A003056).

			Triangle turned on its side begins:
  1  2  2  3  2  4  2  4  3  4  2  6 ...
        1  2  5  6 11 13 17 22 27 29 ...
                 1  2  5 10 15 25 37 ...
                             1  2  5 ...

Alois P. Heinz, Rows n = 1..500, flattened
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

Diagonals give A000005, A002133, A002134, A365630, A365631. Cf. A060043, A060044.

Cf. A116608 (reflected rows). - Alois P. Heinz, Jan 29 2014

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      expand(b(n, i-1) +x*add(b(n-i*j, i-1), j=1..n/i))))
    end:
T:= n->(p->seq(coeff(p, x, degree(p)-k), k=0..degree(p)-1))(b(n$2)):
seq(T(n), n=1..25);  # Alois P. Heinz, Jan 29 2014

Reverse /@ Table[Length /@ Split[ Sort[Map[Length, Split /@ IntegerPartitions[n], {1}]]], {n, 24}] (* Wouter Meeussen, Apr 21 2012, updated by Jean-François Alcover, Jan 29 2014 *)

from math import isqrt
from itertools import count, islice
from sympy.utilities.iterables import partitions
def A060177_gen(): # generator of terms
    return (sum(1 for p in partitions(n) if len(p)==k) for n in count(1) for k in range(isqrt((n<<3)+1)-1>>1,0,-1))
A060177_list = list(islice(A060177_gen(),30)) # Chai Wah Wu, Sep 15 2023

T(n,k) = Partitions of n using only k types of piles. Also, Sum_{k=1..A003056(n)} T(n,k)*k = A000070(n). Also, Sum_{k=1..A003056(n)} T(n,k)*(k-1) = A058884(n). - Naohiro Nomoto, Jan 24 2002

G.f. for k-th diagonal (the k-th row of the sideways triangle shown in the example): Sum_{ m_1 < m_2 < ... < m_k} q^(m_1+m_2+...+m_k)/((1-q^m_1)*(1-q^m_2)*...*(1-q^m_k)) = Sum_n T(n, k)*q^n.

More terms from Naohiro Nomoto, Jan 24 2002

A100073 Number of representations of n as the difference of two positive squares.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1, 2, 1, 0, 2, 1, 1, 0, 1, 2, 2, 0, 2, 1, 1, 0, 2, 2, 1, 0, 1, 1, 3, 0, 1, 3, 1, 0, 2, 1, 1, 0, 2, 2, 2, 0, 1, 2, 1, 0, 3, 2, 2, 0, 1, 1, 2, 0, 1, 3, 1, 0, 3, 1, 2, 0, 1, 3, 2, 0, 1, 2, 2, 0, 2, 2, 1, 0, 2, 1, 2, 0, 2, 4, 1, 0, 3, 1, 1, 0, 1, 2, 4

Offset: 1

T. D. Noe, Nov 02 2004

Note that for odd n, a(n) = 1 iff n is a prime, or a prime squared.
A decomposition n = a^2 - b^2 = (a-b)(a+b) = d*(n/d) is given for each divisor d less than (as to exclude b = 0) but having the same parity as n/d. For even n this implies that d and n/d must be even, i.e., 4 | n. This leads to the given formula, a(n) = floor(numdiv(n)/2) for odd n, floor(numdiv(n/4)/2) for n = 4k, 0 else. - M. F. Hasler, Jul 10 2018
a(n) is the number of self-conjugate partitions of n into parts of 2 different sizes, i.e., the order of the set of partitions obtained by the intersection of the partitions in A000700 and A002133. See A270060. - R. J. Mathar, Jun 15 2022

			a(15) = 2 because 15 = 16 - 1 = 64 - 49.

Robert Israel, Table of n, a(n) for n = 1..10000
A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 5.
Christian Aebi and Grant Cairns, Lattice equable quadrilaterals III: tangential and extangential cases, Integers (2023) Vol. 23, #A48.

Cf. A056924 (number of divisors of n that are less than sqrt(n)), A016825 (numbers not the difference of two squares), A034178 (number of representations of n as the difference of two squares).

Cf. A000700, A002133.

A100073:= proc(n)
  if n::odd then floor(numtheory:-tau(n)/2)
  elif (n/2)::odd then 0
  else floor(numtheory:-tau(n/4)/2)
  fi
end proc:
map(A100073, [$1..200]); # Robert Israel, Jul 10 2018

nn=150; a=Table[0, {nn}]; Do[y=x-1; While[d=x^2-y^2; d<=nn&&y>0, a[[d]]++; y-- ], {x, 1+nn/2}]; a

a(n) = if (n % 2, ceil((numdiv(n)-1)/2), if (!(n%4),  ceil((numdiv(n/4)-1)/2), 0)); \\ Michel Marcus, Mar 07 2016

A100073(n)=if(bittest(n,0),numdiv(n)\2,!bittest(n,1),numdiv(n\4)\2) \\ or shorter: a(n)=if(n%4!=2,numdiv(n\4^!(n%2))\2) \\ - M. F. Hasler, Jul 10 2018

a(n) = A056924(n) for odd n, a(n) = A056924(n/4) if 4|n, otherwise a(n) = 0.

A309058 Partitions of n with parts having at most 3 distinct magnitudes.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 54, 72, 91, 115, 145, 177, 215, 258, 308, 364, 424, 491, 568, 651, 742, 838, 940, 1065, 1181, 1320, 1454, 1619, 1757, 1957, 2124, 2329, 2510, 2763, 2934, 3244, 3432, 3752, 3964, 4329, 4531, 4965, 5179, 5627, 5872, 6391, 6577, 7178, 7405

Offset: 0

Nathan McNew, Jul 09 2019

nonn

Partitions whose Ferrers diagrams do not contain the pattern 4321 under removal of rows and columns (as defined by Bloom and Saracino).

			a(10) = 41 because all of the 42 integer partitions of 10 count (i.e., 10 = 10, 10 = 9+1 = 8+1+1, etc.), except the partition 10 = 4+3+2+1.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
J. Bloom and D. Saracino, Rook and Wilf equivalence of integer partitions, arXiv:1808.04238 [math.CO], 2018.
J. Bloom and D. Saracino, Rook and Wilf equivalence of integer partitions, European J. Combin., 71 (2018), 246-267.

Cf. A265250 (partitions of n with parts having at most 2 distinct magnitudes). Sum of A002134, A002133 and A000005.

Cf. A116608.

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
      `if`(t=1, `if`(irem(n, i)=0, 1, 0)+b(n, i-1, t),
       add(b(n-i*j, i-1, t-`if`(j=0, 0, 1)), j=0..n/i))))
    end:
a:= n-> b(n$2, 3):
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 11 2019

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1, 0, If[t == 1, If[Mod[n, i] == 0, 1, 0] + b[n, i - 1, t], Sum[b[n - i*j, i - 1, t - If[j == 0, 0, 1]], {j, 0, n/i}]]]];
a[n_] := b[n, n, 3];
a /@ Range[0, 100] (* Jean-François Alcover, Feb 27 2020, after Alois P. Heinz *)

G.f.: Sum_{i>=1} x^i/(1-x^i) + Sum_{j=1..i-1} x^(i+j)/((1-x^i)*(1-x^j)) + Sum_{k=1..j-1} x^(i+j+k)/((1-x^i)*(1-x^j)*(1-x^k)).

a(n) = Sum_{k=0..3} A116608(n,k). - Alois P. Heinz, Jul 11 2019

A059820 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^3 *Product_{i=1..t} (1-x^i) ).

Original entry on oeis.org

0, 1, 4, 9, 19, 30, 52, 70, 107, 136, 191, 226, 314, 352, 463, 523, 664, 717, 919, 964, 1205, 1282, 1546, 1603, 1992, 2009, 2414, 2504, 2958, 2974, 3606, 3553, 4223, 4273, 4936, 4912, 5885, 5685, 6634, 6654, 7664, 7454, 8822, 8454, 9845

Offset: 0

N. J. A. Sloane, Feb 24 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.4).

Cf. A000005 (k=1), A059819 (k=2), A059820 (k=3), A059821(k=4), A059822 (k=5), A059823 (k=6), A059824 (k=7), A059825 (k=8).
Cf. A002133, A065608, A191822, A191832.
Cf. A000203, A001157, A055507, A191829 (Andrews's D_{0,0,0}(n)), A191831 (Andrews's D_{0,1}(n)).

Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=3

D(x, y, n) = sum(k=1, n-1, sigma(k, x)*sigma(n-k, y));
D000(n) = sum(k=1, n-1, sigma(k, 0)*D(0, 0, n-k));
a(n) = if(n==0, 0, (3*D(0, 0, n)+3*D(0, 1, n)+D000(n)+2*sigma(n, 0)+3*sigma(n)+sigma(n, 2))/6); \\ Seiichi Manyama, Jul 26 2024

a(n) = ( 3*A055507(n-1) + 3*A191831(n) + A191829(n) + 2*sigma_0(n) + 3*sigma(n) + sigma_2(n) )/6. - Seiichi Manyama, Jul 26 2024

A365676 Triangle read by rows: T(n, k) is the number of partitions of n having exactly k distinct part sizes, for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 3, 2, 0, 0, 0, 2, 5, 0, 0, 0, 0, 4, 6, 1, 0, 0, 0, 0, 2, 11, 2, 0, 0, 0, 0, 0, 4, 13, 5, 0, 0, 0, 0, 0, 0, 3, 17, 10, 0, 0, 0, 0, 0, 0, 0, 4, 22, 15, 1, 0, 0, 0, 0, 0, 0, 0, 2, 27, 25, 2, 0, 0, 0, 0, 0, 0, 0, 0, 6, 29, 37, 5, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Peter Luschny, Sep 15 2023

			Triangle T(n, k) starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 2,  0;
  [3] 0, 2,  1,  0;
  [4] 0, 3,  2,  0, 0;
  [5] 0, 2,  5,  0, 0, 0;
  [6] 0, 4,  6,  1, 0, 0, 0;
  [7] 0, 2, 11,  2, 0, 0, 0, 0;
  [8] 0, 4, 13,  5, 0, 0, 0, 0, 0;
  [9] 0, 3, 17, 10, 0, 0, 0, 0, 0, 0;

P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

Variants: A116608 (nonzero terms), A060177.

Cf. A000041 (row sums), A000005 (T(n,1)), A002133 (T(n,2)), A002134 (T(n,3)), A365630 (T(n,4)), A365631 (T(n,5)).

P := proc(n, k, r) option remember; local j;  # after Amir Livne Bar-on
  if n = 0 then return ifelse(k = 0, 1, 0) fi;
  if k = 0 or r = 0 then return 0 fi;
  add(P(n - r * j, k - 1, r - 1), j = 1..iquo(n, r)) + P(n, k, r - 1) end:
A365676row := n -> local k; seq(P(n, k, n), k = 0..n):
seq(print(A365676row(n)), n = 0..9);
# Using the generating function:
p := product(1 + t*x^j/(1 - x^j), j = 1..20):
ser := series(p, x, 20):
seq(seq(coeff(coeff(ser, x, n), t, k), k = 0..n), n = 0..9);

P[n_, k_, r_] := P[n, k, r] = Which[n == 0, If[k == 0, 1, 0], k == 0 || r == 0, 0, True, Sum[P[n-r*j, k-1, r-1], {j, 1, Quotient[n, r]}]+P[n, k, r-1]]; A365676row[n_] := Table[P[n, k, n], {k, 0, n}]; Table[A365676row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Oct 21 2023, from 1st Maple program *)

T(n,k) = my(nb=0); forpart(p=n, if (#Set(p) == k, nb++)); nb; \\ Michel Marcus, Sep 17 2023

# after Amir Livne Bar-on
from functools import cache
@cache
def P(n: int, k: int, r: int) -> int:
    if n == 0: return 1 if k == 0 else 0
    if k == 0 or r == 0: return 0
    return sum(P(n - r * j, k - 1, r - 1)
               for j in range(1, n // r + 1)) + P(n, k, r - 1)
def A365676Row(n) -> list[int]:
    return [P(n, k, n) for k in range(n + 1)]
for n in range(10): print(A365676Row(n))

T(n, k) = [t^k][x^n] Product_{j>=1} (1 + t*x^j / (1 - x^j)).

T(n, k) = 0 for n>0 and k=0. T(n,k) = 0 for k > floor([sqrt(1+8n)-1]/2). - Chai Wah Wu, Sep 15 2023

cp's OEIS Frontend

A216669 Total number of parts in all partitions of n into 2 sizes of parts (A002133).

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A045845 Duplicate of A002133.

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A238279 Triangle read by rows: T(n,k) is the number of compositions of n into nonzero parts with k parts directly followed by a different part, n>=0, 0<=k<=A004523(n-1).

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A116608 Triangle read by rows: T(n,k) is number of partitions of n having k distinct parts (n>=1, k>=1).

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A238130 Triangle read by rows: T(n,k) is the number of compositions into nonzero parts with k parts directly followed by a different part, n>=0, 0<=k<=n.

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A060177 Triangle of generalized sum of divisors function, read by rows.

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A100073 Number of representations of n as the difference of two positive squares.

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A059820 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^3 *Product_{i=1..t} (1-x^i) ).